Translation:He loves me because I love him.
It is true that 'eftersom' can be translated to both 'because' and 'since'. However the word 'since' has (at least?) two meanings, 'the because of-meaning' and the 'time since-meaning'. The swedish word 'eftersom' can only mean the former (the because of-meaning).
The word 'since' (time since-meaning) is translated as 'sedan'.
It is 'sedan', see my post on the top level question. 'Han älskade mig sedan han var ett barn' would be the direct translation...but it sounds a bit wierd to me since it suggests that the child stopped loving [you].
People would propbably say 'Han har älskat mig sedan han var ett barn' which means that it [love] is still going on....but this is nuances.
I don't feel any real difference in meaning between the two, but därför att doesn't work at the start of a sentence, or at least that sounds very clunky.
Eftersom nästan alla var sjuka ställdes mötet in 'Since almost everyone was ill, the meeting was cancelled' – this would sound odd with därför att instead.
The algorithm says you can have one letter wrong as long as it doesn't turn into another word, so you shouldn't be failed for just älsker since there's no such word. But one thing I know happens is that it sometimes doesn't highlight all errors, so it might fail you for one error, but only underline/mark as red/whatever some typo instead.
So that could be one explanation, but there can sometimes other glitches too. Anyway it shouldn't fail you for just that, in theory.